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Evaluate the limit $$\lim_{x\to\theta}\frac{x\cos\theta - \theta\cos x}{x - \theta}.$$
Find the derivative of \( y = e^{3x - 1} \) by the definition method.
Verify Rolle's theorem for the function \(f(x) = \sin x,\;x\in[0,\pi]\) and find the point on the curve where the tangent is parallel to the \(x\)-axis.
Find maximum and minimum values of the function $$f(x) = x^3 + 6x^2 + 9x - 2.$$ Also find the point of inflection, if any.
Evaluate the integral
a. \(\displaystyle \int \frac{2x + 5}{\sqrt{x^2 + 5x}}\,dx\) b. \(\displaystyle \int e^{2x}\,\log x\,dx\)
Evaluate $$\int_{0}^{2}\sqrt{1 + x^3}\,dx$$ by using Simpson's \(\tfrac{1}{3}\)rule, taking \(n = 4\).
Solve the differential equation: $$2\frac{dy}{dx} = \frac{2y}{x} + \frac{y^2}{x^2}.$$
Attempt any TWO questions
[2x10=20]Define pivot element, pivot column, and consistency in a system of equations. Using the simplex method, maximize $$F = 5x - 3y$$ subject to \[ 3x + 2y \le = 6,\\ -x + 3y \ge = -4,\\ x \ge 0,\quad y \ge = 0. \]
a. Verify Lagrange’s mean value theorem for the function $$f(x) = \sqrt{x - 1},\quad x \in [1,3].$$ b. Solve the differential equation $$x\,y\frac{dy}{dx} = x^2 + y^2.$$
Compute the approximate value of the integrate $$\int \frac{1}{1 + x^2}\,dx$$ by using the composite trapezoidal rule with three points, and compare the result with the actual value. Determine the error formula and numerically verify an upper bound on it.